The estimate (REF ) is obtained gathering (REF ), (), (REF ), () and Lemma REF -(iii). The expression (REF ) follows from the Fourier representation (). The estimates (REF ) correspond to ()-(). The estimates (REF ) and (REF ) are obtained by the same method as Lemma [1]} by using a nice duality representations of \({B}_{\perp }^{\pm 1}\)  together with (), (), (), () and () In particular, the reversibility property of \({R}_0\)  is a consequence of (), (), (), () and the reversibility property of \(\mathbf {R}_{\varepsilon r}\) . To prove (REF ), we use Lemma REF -(ii) together with (), (REF ) and
\(\forall \,0<\lambda _0<\lambda _1,\quad \sup _{j\in \mathbb {Z}}\Big (|j|\max _{k\in \llbracket 0,q\rrbracket }\big \Vert (I_jK_j)^{(k)}\big \Vert _{L^{\infty }([\lambda _0,\lambda _1])}\Big )\leqslant C(\lambda _0,\lambda _1).\) 
